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Properties

Label 16.0.78085528018...8576.1
Degree $16$
Signature $[0, 8]$
Discriminant $2^{44}\cdot 3^{12}\cdot 17^{4}$
Root discriminant $31.14$
Ramified primes $2, 3, 17$
Class number $20$ (GRH)
Class group $[2, 10]$ (GRH)
Galois group $C_2^2:D_4$ (as 16T43)

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Show commands for: Magma / SageMath / Pari/GP

magma: R<x> := PolynomialRing(Rationals()); K<a> := NumberField(R![1633, -1004, 2500, -3372, 2904, -2288, 2400, -2028, 1537, -1112, 724, -388, 186, -84, 32, -8, 1]);
 
sage: x = polygen(QQ); K.<a> = NumberField(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 186*x^12 - 388*x^11 + 724*x^10 - 1112*x^9 + 1537*x^8 - 2028*x^7 + 2400*x^6 - 2288*x^5 + 2904*x^4 - 3372*x^3 + 2500*x^2 - 1004*x + 1633)
 
gp: K = bnfinit(x^16 - 8*x^15 + 32*x^14 - 84*x^13 + 186*x^12 - 388*x^11 + 724*x^10 - 1112*x^9 + 1537*x^8 - 2028*x^7 + 2400*x^6 - 2288*x^5 + 2904*x^4 - 3372*x^3 + 2500*x^2 - 1004*x + 1633, 1)
 

Normalized defining polynomial

\( x^{16} - 8 x^{15} + 32 x^{14} - 84 x^{13} + 186 x^{12} - 388 x^{11} + 724 x^{10} - 1112 x^{9} + 1537 x^{8} - 2028 x^{7} + 2400 x^{6} - 2288 x^{5} + 2904 x^{4} - 3372 x^{3} + 2500 x^{2} - 1004 x + 1633 \)

magma: DefiningPolynomial(K);
 
sage: K.defining_polynomial()
 
gp: K.pol
 

Invariants

Degree:  $16$
magma: Degree(K);
 
sage: K.degree()
 
gp: poldegree(K.pol)
 
Signature:  $[0, 8]$
magma: Signature(K);
 
sage: K.signature()
 
gp: K.sign
 
Discriminant:  \(780855280180961608728576=2^{44}\cdot 3^{12}\cdot 17^{4}\)
magma: Discriminant(Integers(K));
 
sage: K.disc()
 
gp: K.disc
 
Root discriminant:  $31.14$
magma: Abs(Discriminant(Integers(K)))^(1/Degree(K));
 
sage: (K.disc().abs())^(1./K.degree())
 
gp: abs(K.disc)^(1/poldegree(K.pol))
 
Ramified primes:  $2, 3, 17$
magma: PrimeDivisors(Discriminant(Integers(K)));
 
sage: K.disc().support()
 
gp: factor(abs(K.disc))[,1]~
 
This field is not Galois over $\Q$.
This is a CM field.

Integral basis (with respect to field generator \(a\))

$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $\frac{1}{149} a^{12} - \frac{6}{149} a^{11} - \frac{9}{149} a^{10} - \frac{49}{149} a^{9} - \frac{57}{149} a^{8} + \frac{9}{149} a^{7} + \frac{36}{149} a^{6} - \frac{26}{149} a^{5} - \frac{45}{149} a^{4} + \frac{35}{149} a^{3} - \frac{8}{149} a^{2} - \frac{30}{149} a + \frac{15}{149}$, $\frac{1}{149} a^{13} - \frac{45}{149} a^{11} + \frac{46}{149} a^{10} - \frac{53}{149} a^{9} - \frac{35}{149} a^{8} - \frac{59}{149} a^{7} + \frac{41}{149} a^{6} - \frac{52}{149} a^{5} + \frac{63}{149} a^{4} + \frac{53}{149} a^{3} + \frac{71}{149} a^{2} - \frac{16}{149} a - \frac{59}{149}$, $\frac{1}{63018209} a^{14} - \frac{7}{63018209} a^{13} + \frac{102711}{63018209} a^{12} - \frac{616175}{63018209} a^{11} - \frac{19546305}{63018209} a^{10} - \frac{22656789}{63018209} a^{9} - \frac{25843468}{63018209} a^{8} - \frac{19535869}{63018209} a^{7} - \frac{30126060}{63018209} a^{6} - \frac{30456302}{63018209} a^{5} + \frac{13904040}{63018209} a^{4} + \frac{73204}{63018209} a^{3} + \frac{3440329}{63018209} a^{2} + \frac{5224272}{63018209} a + \frac{18649039}{63018209}$, $\frac{1}{618649757753} a^{15} + \frac{4901}{618649757753} a^{14} - \frac{534106128}{618649757753} a^{13} + \frac{1899617654}{618649757753} a^{12} + \frac{216985400871}{618649757753} a^{11} + \frac{289633043856}{618649757753} a^{10} - \frac{110488600658}{618649757753} a^{9} + \frac{71514432181}{618649757753} a^{8} + \frac{57568888378}{618649757753} a^{7} + \frac{210622701042}{618649757753} a^{6} - \frac{205440598703}{618649757753} a^{5} - \frac{226429929055}{618649757753} a^{4} - \frac{106317797574}{618649757753} a^{3} - \frac{68440525392}{618649757753} a^{2} + \frac{35741866514}{618649757753} a - \frac{22758479490}{618649757753}$

magma: IntegralBasis(K);
 
sage: K.integral_basis()
 
gp: K.zk
 

Class group and class number

$C_{2}\times C_{10}$, which has order $20$ (assuming GRH)

magma: ClassGroup(K);
 
sage: K.class_group().invariants()
 
gp: K.clgp
 

Unit group

magma: UK, f := UnitGroup(K);
 
sage: UK = K.unit_group()
 
Rank:  $7$
magma: UnitRank(K);
 
sage: UK.rank()
 
gp: K.fu
 
Torsion generator:  \( -1 \) (order $2$)
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
 
sage: UK.torsion_generator()
 
gp: K.tu[2]
 
Fundamental units:  Units are too long to display, but can be downloaded with other data for this field from 'Stored data to gp' link to the right (assuming GRH)
magma: [K!f(g): g in Generators(UK)];
 
sage: UK.fundamental_units()
 
gp: K.fu
 
Regulator:  \( 9072.35800888 \) (assuming GRH)
magma: Regulator(K);
 
sage: K.regulator()
 
gp: K.reg
 

Galois group

$C_2^2:D_4$ (as 16T43):

magma: GaloisGroup(K);
 
sage: K.galois_group(type='pari')
 
gp: polgalois(K.pol)
 
A solvable group of order 32
The 14 conjugacy class representatives for $C_2^2:D_4$
Character table for $C_2^2:D_4$

Intermediate fields

\(\Q(\sqrt{6}) \), \(\Q(\sqrt{3}) \), \(\Q(\sqrt{2}) \), 4.0.27648.1 x2, 4.0.13824.1 x2, 4.4.4352.1, \(\Q(\sqrt{2}, \sqrt{3})\), 4.4.9792.1, 8.8.1534132224.1, 8.0.883660161024.5, 8.0.3057647616.7

Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.

Sibling fields

Galois closure: data not computed
Degree 16 siblings: data not computed

Frobenius cycle types

$p$ 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59
Cycle type R R ${\href{/LocalNumberField/5.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/7.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/11.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/13.2.0.1}{2} }^{8}$ R ${\href{/LocalNumberField/19.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/23.2.0.1}{2} }^{4}{,}\,{\href{/LocalNumberField/23.1.0.1}{1} }^{8}$ ${\href{/LocalNumberField/29.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/31.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/37.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/41.4.0.1}{4} }^{4}$ ${\href{/LocalNumberField/43.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/47.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/53.2.0.1}{2} }^{8}$ ${\href{/LocalNumberField/59.2.0.1}{2} }^{8}$

In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.

magma: p := 7; // to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
magma: idealfactors := Factorization(p*Integers(K)); // get the data
 
magma: [<primefactor[2], Valuation(Norm(primefactor[1]), p)> : primefactor in idealfactors];
 
sage: p = 7; # to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
sage: [(e, pr.norm().valuation(p)) for pr,e in K.factor(p)]
 
gp: p = 7; \\ to obtain a list of $[e_i,f_i]$ for the factorization of the ideal $p\mathcal{O}_K$:
 
gp: idealfactors = idealprimedec(K, p); \\ get the data
 
gp: vector(length(idealfactors), j, [idealfactors[j][3], idealfactors[j][4]])
 

Local algebras for ramified primes

$p$LabelPolynomial $e$ $f$ $c$ Galois group Slope content
2Data not computed
3Data not computed
$17$17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.4.0.1$x^{4} - x + 11$$1$$4$$0$$C_4$$[\ ]^{4}$
17.8.4.1$x^{8} + 6358 x^{4} - 4913 x^{2} + 10106041$$2$$4$$4$$C_4\times C_2$$[\ ]_{2}^{4}$